Calderón Gómez, José E.

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    Imagen de los τ-Productos
    (2019-05-15) Calderón Gómez, José E.; Ortiz Albino, Reyes M.; College of Arts and Sciences - Arts; Ocasio González, Víctor A.; Dziobiak, Stanislaw; Department of Mathematics; Irizarry Hernández, Zollianne
    The theory of τ-factorizations, also known as theory of generalized factorizations, was developed by Anderson and Frazier in 2006. It was the result of a generalization of the comaximal factorizations by McAdams and Swam, replacing the condition of being comaximals to being related on the set of nonzero nonunits elements in the integral domain. Denote D as an integral domain, U(D) as the set of units of D and D# as the set of elements nonzero nonunits of D. The authors considered symmetric relations defined over the nonzero nonunits elements. The usual theory of factorizations came to be a particular case, where the relation used is τ = D# × D#. An expression of the form a = λa1 · · · an, where λ ∈ U(D) and aiτ aj for all 1 ≤ i 6= j ≤ n, is called a τ-factorizarion of a. Each ai is called a τ-factor of a and a is a τ -product of ai. Furthermore, it is possible to obtain particular cases, such as factorizations in irreducibles elements, primals, and others, by taking τ = S × S, where S is the set of irreducible elements or primals respectively. This work studied the relation τR, where R ⊆ D × E, D and E are integral domains, and τ is defined on D#. The relation τR is defined as xτRy, if and only if there exist a, b ∈ D# such that aτ b, aRx, and bRy. That is, τR is “the image of τ with respect to the relation R”. The properties of τR that can be inherited from τ in τR are analyzed. It must be clarified that although the definition is given with respect to the image of a relation, most of the work is focused in different types of functions, such as one to one and surjectives functions, homomorphisms, and others. The principal objective is to provide a way to study τ-factorizations and structural properties using the images of the functions.